Melnikov distance
In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
The Melnikov method is used in many cases to predict the occurrence of chaotic orbits in non-autonomous smooth nonlinear systems under periodic perturbation.
According to the method, it is possible to construct a function called the "Melnikov function" which can be used to predict either regular or chaotic behavior of a dynamical system.
Thus, the Melnikov function will be used to determine a measure of distance between stable and unstable manifolds in the Poincaré map.
Moreover, it was described by several textbooks as Guckenheimer & Holmes,[3] Kuznetsov,[4] S. Wiggins,[5] Awrejcewicz & Holicke[6] and others.
There are many applications for Melnikov distance as it can be used to predict chaotic vibrations.
[7] In this method, critical amplitude is found by setting the distance between homoclinic orbits and stable manifolds equal to zero.
Just like in Guckenheimer & Holmes where they were the first who based on the KAM theorem, determined a set of parameters of relatively weak perturbed Hamiltonian systems of two-degrees-of-freedom, at which homoclinic bifurcation occurred.
Assume that system (1) is smooth on the region of interest,
From this system (3), looking at the phase space in Figure 1, consider the following assumptions To obtain the Melnikov function, some tricks have to be used, for example, to get rid of the time dependence and to gain geometrical advantages new coordinate has to be used
Then, the system (1) could be rewritten in vector form as follows
Hence, looking at Figure 2, the three-dimensional phase space
of the unperturbed system becoming a periodic orbit
of the unperturbed vector field (3) persists as a periodic orbit,
are the trajectories of the unperturbed and perturbed vector fields, respectively.
Expanding in Taylor series the eq.
(6) it will require knowing the solution to the perturbed problem.
The first two terms on the right hand side can be verified to cancel by explicitly evaluating the matrix multiplications and dot products.
Integrating the remaining term, the expression for the original terms does not depend on the solution of the perturbed problem.
The lower integration bound has been chosen to be the time where
the final form for the Melnikov distance is obtained by
From theorem 1 when there is a simple zero of the Melnikov function implies in transversal intersections of the stable
Such tangle is a very complicated structure with the stable and unstable manifolds intersecting an infinite number of times.
Consider a small element of phase volume, departing from the neighborhood of a point near the transversal intersection, along the unstable manifold of a fixed point.
Clearly, when this volume element approaches the hyperbolic fixed point it will be distorted considerably, due to the repetitive infinite intersections and stretching (and folding) associated with the relevant invariant sets.
Therefore, it is reasonably expect that the volume element will undergo an infinite sequence of stretch and fold transformations as the horseshoe map.
is an n-dimensional manifold, has a hyperbolic fixed point
unstable manifold that intersect transversely at some point
is topologically conjugate to a shift on finitely many symbols.
Thus, according to the theorem 2, it implies that the dynamics with a transverse homoclinic point is topologically similar to the horseshoe map and it has the property of sensitivity to initial conditions and hence when the Melnikov distance (10) has a simple zero, it implies that the system is chaotic.
